Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz operators over Banach spaces using DNNs with applications to various PDE solution operators. The goal is to specify DNN width, depth, and the number of training samples needed to guarantee a certain testing error. Under mild assumptions on data distributions or operator structures, our analysis shows that deep operator learning can have a relaxed dependence on the discretization resolution of PDEs and, hence, lessen the curse of dimensionality in many PDE-related problems including elliptic equations, parabolic equations, and Burgers equations. Our results are also applied to give insights about discretization-invariance in operator learning.

翻译：深度神经网络（DNNs）已在众多领域取得显著成功，其在偏微分方程相关问题的应用也迅速发展。本文给出了利用DNNs学习巴拿赫空间上Lipschitz算子泛化误差的估计，并将其应用于多种偏微分方程解算子。旨在明确为保证特定测试误差所需的DNN宽度、深度及训练样本数量。在数据分布或算子结构的温和假设下，我们的分析表明，深度算子学习可放宽对偏微分方程离散化分辨率的依赖，从而在包括椭圆方程、抛物方程及Burgers方程在内的众多偏微分方程相关问题的求解中缓解维数灾难。本文结果亦为算子学习中的离散化不变性提供了见解。